THE SCHUR-HORN THEOREM FOR OPERATORS WITH FINITE SPECTRUM

Abstract

The carpenter problem in the context of II1 factors, formulated by Kadison, asks: Let A subset of M be a masa in a type II1 factor and let E be the normal conditional expectation from M onto A. Then, is it true that for every positive contraction A in A, there is a projection P in M such that E(P) = A? In this note, we show that this is true if A has finite spectrum. We will then use this result to prove an exact Schur-Horn theorem for positive operators with finite spectrum in type II1 factors and an approximate Schur-Horn theorem for general positive operators in type II1 factors.